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On supercompactness and the continuum function
Annals of Pure and Applied Logic 165 (2014) 620–630
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On supercompactness and the continuum function ✩ Brent Cody a,∗,1 , Menachem Magidor b,2 a The Fields Institute for Research in Mathematical Sciences, 222 College Street, Toronto, Ontario M5S 2N2, Canada b Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
a r t i c l e
i n f o
Article history: Received 20 May 2013 Received in revised form 10 September 2013 Accepted 10 September 2013 Available online 7 October 2013 MSC: 03E05 03E10 03E35 03E40 03E50 03E55
a b s t r a c t Given a cardinal κ that is λ-supercompact for some regular cardinal λ κ and assuming GCH, we show that one can force the continuum function to agree with any function F : [κ, λ] ∩ REG → CARD satisfying ∀α, β ∈ dom(F ) α < cf(F (α)) and α < β =⇒ F (α) F (β), while preserving the λ-supercompactness of κ from a hypothesis that is of the weakest possible consistency strength, namely, from the hypothesis that there is an elementary embedding j : V → M with critical point κ such that M λ ⊆ M and j(κ) > F (λ). Our argument extends Woodin’s technique of surgically modifying a generic filter to a new case: Woodin’s key lemma applies when modifications are done on the range of j, whereas our argument uses a new key lemma to handle modifications done off of the range of j on the ghost coordinates. This work answers a question of Friedman and Honzik [5]. We also discuss several related open questions. © 2013 Elsevier B.V. All rights reserved.
Keywords: Supercompact cardinal Continuum function Forcing Large cardinal
1. Introduction The behavior of the continuum function γ → 2γ on the regular cardinals was shown, by Easton, to be highly undetermined by the axioms of ZFC. Easton proved [3] that if F is any function from the regular cardinals to the cardinals satisfying α < cf(F (α)) and α < β =⇒ F (α) F (β), then there is a cofinality-preserving forcing extension in which 2γ = F (γ) for every regular cardinal γ. Large cardinal axioms impose additional restrictions on the continuum function on the regular cardinals. For example, ✩ The first author would like to thank the Fields Institute as well as the University of Prince Edward Island for their support while this work was being carried out. * Corresponding author. E-mail addresses: [email protected] (B. Cody), [email protected] (M. Magidor). URL: http://www.people.vcu.edu/~bmcody/ (B. Cody). 1 Current address: Virginia Commonwealth University, Department of Mathematics and Applied Mathematics, 1015 Floyd Avenue, Richmond, Virginia, 23284, United States. 2 Tel.: +972 2 65 84143.
0168-0072/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.apal.2013.09.001
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if κ is a supercompact cardinal and GCH holds below κ, then GCH holds everywhere. It therefore seems natural to address the question: What functions can be forced to coincide with the continuum function on the regular cardinals while preserving large cardinals? From what hypotheses? In particular, let us consider the following question. Question 1. Given a λ-supercompact cardinal κ where λ κ is a regular cardinal, and assuming GCH, what functions F from the regular cardinals to the cardinals can be forced to equal the continuum function on the interval [κ, λ] while preserving the λ-supercompactness of κ and preserving cardinals? From what hypotheses? Menas [11] proved that, assuming GCH, one can force the continuum function to agree at every regular cardinal with any locally definable 3 function F satisfying the requirements of Easton’s theorem, while preserving all cofinalities and preserving the supercompactness of a cardinal κ. If κ is a Laver-indestructible supercompact cardinal in a model V , as in [9], it easily follows that one can force over this model to achieve any reasonable behavior of the continuum function at and above κ, while preserving the supercompactness of κ. In particular, starting with a Laver-indestructible supercompact cardinal, one can obtain a model with a measurable cardinal at which GCH fails. However, one can also obtain a model with a measurable cardinal at which GCH fails from a much weaker large cardinal assumption. Woodin proved that the existence of a measurable cardinal at which GCH fails is equiconsistent with the existence of an elementary embedding j : V → M with critical point κ such that M κ ⊆ M and j(κ) > κ++ (see [2, Theorem 25.1]). Woodin’s argument illustrates that under certain conditions, one may perform a type of surgical modification on a generic filter g to obtain g ∗ in order to meet the lifting criterion, j G ⊆ g ∗ , and such that g ∗ remains generic. In Woodin’s proof, the modifications made to g in order to obtain g ∗ only occur on the range of j, and his key lemma shows that such changes are relatively mild in the sense that for a given condition p ∈ g, the set over which modifications are made to obtain p∗ ∈ g ∗ has size at most κ. Hamkins showed [7] that Woodin’s method could be applied to obtain an indestructibility theorem for tall cardinals. The first author extended Woodin’s surgery method to the case of partially supercompact cardinals in [1]. It is shown in [1] that the existence of a λ-supercompact cardinal κ such that 2λ λ++ is equiconsistent with the following hypothesis. (∗) There is an elementary embedding j : V → M with critical point κ such that M λ ⊆ M and j(κ) > λ++ . The method used in [1] is to, after a suitable preparatory iteration, blow up the size of the powerset of κ using Cohen forcing, in order to achieve 2κ = λ++ and then use Woodin’s method of surgery to lift the elementary embedding. Thus, one obtains a model in which κ is λ-supercompact and GCH fails at λ, because 2κ = λ++ . Answering a question posed in [1], Friedman and Honzik [5] used a variant of Sacks forcing for uncountable cardinals to show, from the hypothesis (∗), one can obtain a forcing extension in which κ is λ-supercompact, GCH holds on [κ, λ), and 2λ λ++ . The methods of both [1] and [5] leave open the following question, which appears in [5]. Assuming GCH and (∗), where κ < γ < λ are regular cardinals, is there a cofinality-preserving forcing extension in which κ remains λ-supercompact, GCH holds on the interval [κ, γ), and 2γ = λ++ ? In this article we answer this question, and indeed, provide a full answer to Question 1, by proving the following theorem. Theorem 1. Suppose GCH holds, κ < λ are regular cardinals, and F : [κ, λ] ∩ REG → CARD is a function such that ∀α, β ∈ dom(F ) α < cf(F (α)) and α < β =⇒ F (α) F (β). If there is an elementary embedding 3 A function F is locally definable if there is a sentence ψ, true in V , and a formula ϕ(x, y) such that for all cardinals γ, if Hγ |= ψ, then F has a closure point at γ and for all α, β < γ, we have F (α) = β ↔ Hγ |= ϕ(α, β).
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j : V → M with critical point κ such that M λ ⊆ M and j(κ) > F (λ), then there is a cofinality-preserving forcing extension in which κ remains λ-supercompact and 2γ = F (γ) for every regular cardinal γ ∈ [κ, λ]. The stage κ forcing Q in our proof will be an Easton-support product forcing for adding F (γ)-subsets to γ for each regular cardinal γ ∈ [κ, λ]. Since conditions p in this forcing will have size greater than the critical point of the embedding j under consideration, it follows that j(p) = j p and hence j(p) will have ghost-coordinates. Part of the modification we will perform on a certain generic filter g to obtain g ∗ with j H ⊆ g ∗ will occur off of the range of j on these ghost coordinates. Hence, Woodin’s key lemma does not apply. We will formulate and prove a new key lemma (Lemma 4 below) that allows much of the rest of the argument to be carried out as before. Essentially, our new key lemma shows that the set over which modifications are made to a condition p ∈ g to obtain p∗ ∈ g ∗ can be broken up into λ pieces, each of which is in the relevant model. The hypothesis in Theorem 1 is of optimal consistency strength for the simple reason that if j : V → M witnesses that κ is λ-supercompact, then M < j(κ). 2 λ 2λ
In Section 4 we discuss an easy corollary of Theorem 1 and an open question both addressing the case when λ is a singular cardinal. 2. Preliminaries 2.1. Background material We assume familiarity with the large cardinal notions of measurable and supercompact cardinals as well as with the characterization of these notions in terms of the existence of nontrivial elementary embeddings from the universe V into transitive inner models M ⊆ V . Some familiarity with the techniques of lifting large cardinal embeddings to forcing extensions is also assumed (see [2, Sections 8 and 9] and [8, Chapter 21]). 2.2. Supercompactness with tallness We say that a cardinal κ is θ-tall if θ > κ is an ordinal and there is an elementary embedding j : V → M with critical point κ such that j(κ) > θ and M κ ⊆ M . We say that κ is λ-supercompact with tallness θ if κ λ are cardinals, θ > λ is an ordinal, and there is a j : V → M with critical point κ such that M λ ⊆ M and j(κ) > θ. Such cardinals have been studied by Hamkins [7], Cody [1], Friedman and Honzik [5], and others. The next lemma appears in [1], and is easy to verify by factoring an elementary embedding through the ultrapower by an extender. Lemma 1. If κ is λ-supercompact with tallness θ then there is an embedding j : V → M witnessing this such that M = j(h) j λ, α | h : Pκ λ × κ → V & α < θλ & h ∈ V .
2.3. Easton’s product forcing Let us quickly review Easton’s product forcing and fix some notation. Suppose κ and λ are regular cardinals and F : [κ, λ] ∩ REG → CARD is a function satisfying Easton’s requirements; that is, for all regular cardinals α, β ∈ [κ, λ] one has
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α < cf F (α) and
(E1)
α < β =⇒ F (α) F (β).
(E2)
γ Let QF [κ,λ] denote the Easton-support product of Cohen forcing to achieve 2 = F (γ) for every regular F γ ∈ [κ, λ], by forcing over a model of GCH. We can regard conditions p ∈ Q[κ,λ] as functions satisfying the following.
• Every element in dom(p) is of the form (γ, α, β) where γ ∈ [κ, λ] is a regular cardinal, α < γ, and β < F (γ). • (Easton support) For each regular cardinal γ ∈ [κ, λ] we have (δ, α, β) ∈ dom(p) δ γ < γ.
• ran(p) ⊆ {0, 1}. Let dom(QF [κ,λ] ) :=
{dom(p) | p ∈ QF [κ,λ] } and notice that
dom QF [κ,λ] = (δ, α, β) δ ∈ [κ, λ] ∩ REG, α < δ, and β < F (δ)
By [3] we obtain the following. γ Lemma 2. Assuming GCH, forcing with the poset QF [κ,λ] preserves all cofinalities and achieves 2 = F (γ) for every regular cardinal γ ∈ [κ, λ] while preserving GCH otherwise.
One may also see [8, Theorem 15.18] for further details regarding the partial order QF [κ,λ] and Lemma 2. 3. Proof of the main theorem The next lemma will allow us to define the forcing iteration we will use to prove Theorem 1. Notice that if GCH holds and κ, λ, F , and j are as in the statement of Theorem 1 then F (λ)λ = F (λ) since λ < cf(F (λ)). Thus, under the hypothesis of Theorem 1, it follows from Lemma 1 that we may assume without loss of generality that j : V → M is such that M = j(h) j λ, α h : Pκ λ × κ → V & α < F (λ) & h ∈ V .
(3.1)
Lemma 3. Assume GCH and that κ, λ, F , and j : V → M are as in the hypothesis of Theorem 1. Furthermore, assume that M is as in (3.1). Then there is a cofinality-preserving forcing extension V [G] such that the following hold. (1) GCH holds in V [G]. (2) There is a partial function f from κ to Vκ in V [G] such that j lifts to j : V [G] → M [j(G)] and j(f )(κ) = F . (3) M [j(G)] = {j(h)(j λ, α) | h : (Pκ λ)V × κ → V & α < F (λ) & h ∈ V [G]}. M Proof. Fix x = xξ | ξ < κ , a well-ordering of Vκ . Then j( x) = zξ | ξ < j(κ) is a well-ordering of Vj(κ) . λ M Since M ⊆ M it is clear that F ∈ M and hence F ∈ Vj(κ) . Fix α < j(κ) such that F = zα . Let Fκ be Woodin’s poset for adding a partial function from κ to κ, defined as follows. Conditions in Fκ are partial functions p ⊆ κ × κ satisfying the following conditions.
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• γ ∈ dom(p) =⇒ γ < κ is an inaccessible cardinal and p γ ⊆ γ. • For every inaccessible cardinal μ < κ one has |p μ| < μ. The ordering on Fκ is defined by p p if and only if p ⊇ p . For a proof that under GCH the poset Fκ preserves all cofinalities and does not disturb the continuum function, see [6, Theorem 1.3]. Let G be V -generic for Fκ and let f˜ = G. Clearly f˜ is a partial function from κ to κ. Let us show that in M the poset j(Fκ ) factors below a condition p0 as j(Fκ )/p0 ∼ = Fκ × Ftail such that Ftail is F (λ)-closed in M . Notice that {(κ, α)} is a condition in Fκ and that the forcing j(Fκ ) can be factored below {(κ, α)} as j(Fκ )/{(κ, α)} ∼ = Fκ ×F[γ0 ,j(κ)) where γ0 denotes the least M -inaccessible cardinal greater than α, and F[γ0 ,j(κ)) is a γ0 -closed poset in M . If γ0 F (λ) then we can take p0 = {(κ, α)} and Ftail = F[γ0 ,j(κ)) . Assuming γ0 < F (λ), let p0 = {(κ, α), (γ0 , F (λ))} and factor j(Fκ ) below the condition p0 = {(κ, α), (γ0 , F (λ))} to obtain j(Fκ )/ (κ, α), γ0 , F (λ) ∼ = Fκ × F[γ1 ,j(κ))
where γ1 is the least M -inaccessible cardinal greater than F (λ) and F[γ1 ,j(κ)) is γ1 -closed in M . In this case we take Ftail = F[γ1 ,j(κ)) . Now we will show that there is an M -generic filter K for j(Fκ )/p0 ∼ = Fκ × Ftail in V [G] satisfying the lifting criterion j G ⊆ K, and hence j lifts to j : V [G] → M [j(G)] (see [2, Proposition 9.1]) where j(G) is the upward closure of K in j(Fκ ). This will suffice for (2) because p0 ∈ j(G) ensures that j(f˜)(κ) = α, and then working in V [G], one can define a function f with dom(f ) ⊆ κ by f (ξ) = xf˜(ξ) , and this function satisfies j(f )(κ) = zj(f˜)(κ) = zα = F . Let us show that K, as above, can be built in V [G]. First we show that there is an M -generic filter Gtail for Ftail in V . Let X denote the set of all dense subsets of all tails of the forcing Fκ . It follows that X has size at most 2κ = κ+ in V . Since M is as in (3.1), every dense subset D of Ftail in M is represented by a function h : Pκ λ × κ → X as D = j(h)(j λ, α) where α < F (λ) and h ∈ V . Since there are at most <κ (2κ )λ = (2κ )λ = 2λ = λ+ such functions we can enumerate them as hξ | ξ < λ+ ∈ V . Working in V we will build a decreasing sequence pξ | ξ < λ+ with pξ ∈ Ftail such that pξ meets every dense subset of Ftail in M represented by some hζ for ζ ξ. Assume pζ has been constructed for ζ < ξ. At stage ξ < λ, since M λ ⊆ M in V , we may let rξ ∈ Ftail be a lower bound of all previously constructed conditions. Since j(hξ )(j λ, α) | α < F (λ) ∈ M and Ftail is F (λ)-closed in M , it follows that there is a single condition pξ below rξ meeting every dense subset of Ftail in the sequence j(hξ )(j λ, α) | α < F (λ) . Since every dense subset of Ftail in M has a name that is represented by some function on our list hξ | ξ < λ+ , we can use pξ | ξ < λ+ to generate Gtail ∈ V an M -generic filter for Ftail . Now let K = G × Gtail . It is easy to verify that K is M -generic for j(Fκ )/p0 ∼ = Fκ × Ftail . We now show that (3) holds. If x ∈ M [j(G)] then x has a j(Fκ )-name, x˙ ∈ M such that x = x˙ j(G) . Thus we may write x˙ = j(h)(j λ, α) for some h : Pκ λ × κ → V , α < F (λ), and h ∈ V . Working in V [G] define ˜ : (Pκ λ)V × κ → V [G] by letting h(z, ˜ ξ) = h(z, ξ)G whenever h(z, ξ) is an Fκ -name and letting a function h ˜ ˜ ˜ h(z, ξ) = ∅ otherwise. Then h ∈ V [G] and j(h)(j λ, α) = j(h)(j λ, α)j(G) = x˙ j(G) = x. 2 Let us restate the main theorem. Theorem 2. Suppose GCH holds, κ < λ are regular cardinals, and F : [κ, λ] ∩ REG → CARD is a function such that ∀α, β ∈ dom(F ) α < cf(F (α)) and α < β =⇒ F (α) F (β). If there is an elementary embedding j : V → M with critical point κ such that M λ ⊆ M and j(κ) > F (λ), then there is a cofinality-preserving forcing extension in which κ remains λ-supercompact and 2γ = F (γ) for every regular cardinal γ ∈ [κ, λ]. be an Proof. To simplify notation later on, let V denote the model we start with and let j : V → M λ elementary embedding, which is a definable class of V such that (1) M ⊆ M , (2) F is as in the statement
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of the theorem, and (3) j(κ) > F (λ). Since GCH implies λ<κ = λ and F (λ)λ = F (λ), we can further assume without loss of generality that = M j(h) j λ, α h : Pκ λ × κ → V & α < F (λ) & h ∈ V . be the forcing extension of Lemma 3. So [ where Now let V = V [G] j lifts to j : V → M = M j(G)] M = j(h) j λ, α h : (Pκ λ)V × κ & α < F (λ) & h ∈ V ,
and there is a partial function f from κ to Vκ in V such that j(f )(κ) = F . Working in V , define Yf ⊆ dom(f ) to be the set of all γ < κ such that the following properties hold. (1) γ ∈ dom(f ) is an inaccessible cardinal and f γ ⊆ Vγ . (2) For some regular cardinal λγ ∈ [γ, κ), the value of f (γ) is a function from [γ, λγ ] ∩ REG to CARD satisfying the requirements of Easton’s theorem. Clearly, Yf must have measure one with respect to the normal measure U = {X ⊆ κ | κ ∈ j(X)}. ˙ γ ) | γ < κ + 1 be the length κ + 1 Easton-support iteration defined as follows. Let Pκ+1 = (Pγ , Q ˙γ =Q ˙ f (γ) be a Pγ -name for the Easton-support product Qf (γ) as defined in V Pγ • If γ ∈ Yf then let Q [γ,λγ ] [γ,λγ ] f (γ)
(see Section 2 above for a definition of the poset Q[γ,λγ ] ). ˙ γ is a Pγ -name for trivial forcing. • If γ < κ and γ ∈ / Yf , then Q F ˙κ =Q ˙F • For the stage κ forcing, let Q [κ,λ] be a Pκ -name for the Easton-support product Q[κ,λ] as defined Pκ in V . ˙ F . Standard arguments show that Pκ ∗ Q ˙ F preserves cofinalities and Let G ∗ H be V -generic for Pκ ∗ Q [κ,λ] [κ,λ] that in V [G][H] one has 2δ = F (δ) for every regular cardinal δ in [κ, λ]. Our argument follows Woodin’s: we show that there is a further cofinality-preserving forcing extension V [G][H][g0 ] in which κ remains λ-supercompact, and the desired continuum function is preserved on the interval [κ, λ]. 3.1. Lifting the embedding through Pκ ˙ j(f )(κ) ∗ P˙ tail . Since j(f )(κ) = F By elementarity, the forcing j(Pκ ) can be factored in M as j(Pκ ) ∼ = Pκ ∗ Q [κ,λκ ] by Lemma 3(3), and since M λ ⊆ M , it follows that ˙ F ∗ P˙ tail . j(Pκ ) ∼ = Pκ ∗ Q [κ,λ] Since the next stage of nontrivial forcing in j(Pκ ) beyond κ must be beyond F (λ), it follows that Ptail := (P˙ tail )G∗H is F (λ)-closed in M [G][H]. Now we will show that there is an M [G][H]-generic filter for Ptail in V [G][H]. The argument is similar to that in the proof of Lemma 3. For ξ < κ let Xξ denote the collection of all nice Pξ -names for dense subsets of the tail P˙ ξ,κ of the iteration Pκ . It follows that Xξ has size at most 2κ and hence X := {Xξ | ξ < κ} has ˙ F -name for D. size at most 2κ . Suppose D ∈ M [G][H] is a dense subset of Ptail and let D˙ ∈ M be a Pκ ∗ Q [κ,λ] ˙ Without loss of generality we may assume that D ∈ j(X) and hence there is a function h : (P λ)V ×κ → X ˙ D
κ
in V with D˙ = j(h)(j λ, α) for some α < F (λ). In other words, every dense subset of Ptail in M [G][H] ˙ F -name that is represented by a function (Pκ λ)V × κ → X in V . Since there are at most has a Pκ ∗ Q [κ,λ]
(2κ )λ
<κ
= 2λ = λ+ -such functions we can enumerate them as hξ | ξ < λ+ ∈ V . Working in V [G][H] we
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build a decreasing sequence pξ | ξ < λ+ with pξ ∈ Ptail meeting every dense subset of Ptail in M [G][H] as follows. Assume pζ has been constructed for ζ < ξ. At stage ξ < λ, since M [G][H]λ ⊆ M [G][H] in V [G][H], we may let rξ ∈ Ptail be a lower bound of all previously constructed conditions. Since j(hξ )(j λ, α)G∗H | α < F (λ) ∈ M [G][H] and Ptail is F (λ)-closed in M [G][H], it follows that there is a single condition pξ below rξ meeting every dense subset of Ptail in the sequence j(hξ )(j λ, α)G∗H | α < F (λ) . Since every dense subset of Ptail in M [G][H] has a name that represented by some function on our list hξ | ξ < λ+ , we can use pξ | ξ < λ+ to generate an M [G][H]-generic filter, call it Gtail ∈ V [G][H]. Since conditions in Pκ have bounded support, it follows that j G ⊆ G ∗ H ∗ Gtail , and thus we may lift the embedding to j : V [G] → M [j(G)] in V [G][H] where j(G) = G∗H ∗Gtail (as before, see [2, Proposition 9.1]). We have
M j(G) = j(h) j λ, α h : (Pκ λ)V × κ → V [G] & α < F (λ) & h ∈ V [G] .
3.2. Obtaining an M [j(G)]-generic filter for j(QF [κ,λ] ) We will factor the embedding j : V [G] → M [j(G)] through an ultrapower embedding j0 , force over F V [G][H] with j0 (QF [κ,λ] ) to obtain g0 , and then transfer g0 to an M [j(G)]-generic filter g for j(Q[κ,λ] ). F We will also show that forcing with j0 (Q[κ,λ] ) over V [G][H] preserves cardinals and does not disturb the continuum function below, or at, λ. Let X = {j(h)(j λ) | h : (Pκ λ)V → V [G] where h ∈ V [G]}. Then it follows that X ≺ M [j(G)]. Let k : M0 → M [j(G)] be the inverse of the Mostowski collapse π : X → M0 and let j0 : V [G] → M0 be defined by j0 := π ◦ j. It follows that j0 is the ultrapower embedding by the measure U0 := {X ⊆ (Pκ λ)V | j λ ∈ j(X)} where U0 ∈ V [G][H]. It is shown in [10] that the ground model is definable in any set forcing extension. Thus, by elementarity, M0 is of the form M0 [j0 (G)], where M0 ⊆ M0 and j0 (G) ⊆ j0 (Pκ ) ∈ M0 0 is M0 -generic. Furthermore, j0 (G) = G ∗ H0 ∗ G0tail where H0 is M0 [G]-generic for π(QF [κ,λ] ) and Gtail is M [G][H0 ]-generic for the tail of the iteration j0 (Pκ ) above stage κ. The following diagram is commutative. V [G]
j
j0
M [j(G)] k
M0 [j0 (G)] It follows that j0 is a class of V [G][H0 ], M0 [j0 (G)] is closed under λ-sequences in V [G][H0 ], and that j0 (κ) > π(F (λ)) with |j0 (κ)|V = λ+ . Hence the cardinal structures of M0 [j0 (G)] and V [G][H0 ] are identical up to and including λ+ . Furthermore, since j λ ∈ X it follows that X λ ⊆ X in V [G][H] which implies λ+ ⊆ X and crit(k) λ+ . Notice that F ∈ dom(π) = X because F = j(f )(κ) = k(j0 (f ))(κ) = k(j0 (f )(κ)) ∈ ran(k) = π(F ) dom(π). Thus π(QF [κ,λ] ) = Q[κ,λ] , where π(F ) is a function with domain π([κ, λ]) = [κ, λ] satisfying the requirements of Easton’s theorem in M0 [G]. Let g0 be V [G][H]-generic for j0 (QF [κ,λ] ). F The next claim shows that forcing with j0 (Q[κ,λ] ) over V [G][H] preserves cardinals and does not disturb the continuum function below, or at, λ. ++ -c.c. in V [G][H]. Claim 1. j0 (QF [κ,λ] ) is λ-distributive and λ
Proof. First we will demonstrate the distributivity. By elementarity, j0 (QF [κ,λ] ) is λ-closed in M0 [j0 (G)]. Since G0tail ∈ V [G][H0 ], it follows that j0 (QF ) ∈ V [G][H ] and since M [j 0 0 0 (G)] is closed under λ-sequences [κ,λ] F in V [G][H0 ], it follows that j0 (Q[κ,λ] ) is λ-closed in V [G][H0 ]. Notice that H0 is V [G]-generic for π(QF [κ,λ] ) = π(F ) V [G] F + and that the quotient forcing Q[κ,λ] /H0 is λ -c.c. in V [G][H0 ] Q[κ,λ] = γ∈[κ,λ]∩REG Add(γ, π(F (γ)))
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∗
∗ since it is isomorphic to QF [κ,λ] in that model, where F is a function with domain REG ∩ [κ, λ] such that ∗ + ∗ F (μ) = 0 if F (μ) λ and F (μ) = F (μ) otherwise. Easton’s Lemma states that λ-closed forcing remains λ-distributive after λ+ -c.c. forcing. Hence j0 (QF [κ,λ] ) is λ-distributive in V [G][H]. F ++ Now let us show that j0 (Q[κ,λ] ) is λ -c.c. in V [G][H]. Each p ∈ j0 (QF [κ,λ] ) can be written as p = V F j (h )(j λ) for some function h : (P λ) → Q with h ∈ V [G]. For each p ∈ j (QF ) the domain 0
p
0
p
κ
[κ,λ]
p
0
[κ,λ]
of hp has size λ in V [G], and thus hp leads to hp : λ → QF [κ,λ] , which can be viewed as a condition in the F full support product of λ copies of Q[κ,λ] taken in V [G]. Let us denote this product by QF [κ,λ] . We will show ++ F ++ that j0 (QF ) is λ -c.c. in V [G][H] by arguing that Q is λ -c.c. in V [G][H] and that an antichain [κ,λ] [κ,λ] F ++ F ++ in V [G][H] would lead to an antichain of Q[κ,λ] of size λ in V [G][H]. of j0 (Q[κ,λ] ) of size λ F ++ We now briefly describe how to prove that Q[κ,λ] is λ -c.c. in V [G][H]. An easy delta-system argument ++ shows that QF -c.c. in V [G]. Suppose A is an antichain of size δ of QF [κ,λ] is λ [κ,λ] in V [G][H]. We will show F F F ∼ that A leads to an antichain of Q[κ,λ] = Q[κ,λ] × Q[κ,λ] in V [G] of size δ. Let ˙ ˙ q A˙ is an antichain of QF [κ,λ] and f : δ → A is bijective ˙ ˙ where q ∈ QF ˇα where pα ∈ QF [κ,λ] and AH = A. For each α < δ let qα q be such that qα f (α) = p [κ,λ] . F F F We have Q[κ,λ] ∼ = Q[κ,λ] × Q[κ,λ] in V [G], and it is easy to check that W := {(qα , pα ) | α < δ} is an antichain F F ++ of QF -c.c. in V [G][H]. [κ,λ] × Q[κ,λ] in V [G] of size δ. This shows that Q[κ,λ] is λ It remains to show that an antichain of j0 (QF ) in V [G][H] with size λ++ would lead to an antichain [κ,λ] F ++ of Q[κ,λ] in V [G][H] of size λ . Suppose A is an antichain of j0 (QF [κ,λ] ) with size δ in V [G][H]. Each p ∈ A ˜
is of the form j0 (hp )(j λ) where hp : (Pκ λ)V → QF [κ,λ] . As mentioned above, each hp leads to a condition F hp ∈ Q[κ,λ] . It is easy to check that A := {hp | p ∈ A} is an antichain of QF [κ,λ] . 2 Let us now show that g0 can be transferred along k to an M [j(G)]-generic filter for j(QF [κ,λ] ). F Claim 2. k g0 ⊆ j(QF [κ,λ] ) generates an M [j(G)]-generic filter g for j(Q[κ,λ] ). Proof. Suppose D ∈ M [j(G)] is an open dense subset of j(QF [κ,λ] ) and let D = j(h)(j λ, α) for some h ∈ V [G] with dom(h) = (Pκ λ)V × κ and α < F (λ). Without loss of generality, let us assume that every element of the range of h is a dense subset of QF [κ,λ] in V [G]. We have D = j(h)(j λ, α) = k(j0 (h))(j λ, α). Since π(F (λ)) < j0 (κ) we may define a function h ∈ M0 [j0 (G)] with dom( h) = π(F (λ)) by h(ξ) = j0 (h)(j0 λ, ξ). Then dom(k( h)) = k(π(F (λ))) = F (λ) and since the critical point of k is greater than h)(α) = k(j0 (h))(k(j λ), α) = j(h)(j λ, α) = D. Now the range of h is a collection of λ we have k( 0
F π(F (λ)) open dense subsets of j0 (QF [κ,λ] ). Since j0 (Q[κ,λ] ) is π(F (λ))-distributive in M0 [j0 (G)], one sees
= ran( h) is an open dense subset of j0 (QF that D [κ,λ] ). Hence there is a condition p ∈ g0 ∩ D and by ⊆ D. 2 elementarity, k(p) ∈ k g0 ∩ k(D)
3.3. Performing surgery ∗ ∗ With the M [j(G)]-generic filter g for j(QF [κ,λ] ) in hand, we surgically modify g to obtain g with j H ⊆ g , and then argue that g ∗ remains an M [j(G)]-generic filter for j(QF [κ,λ] ). ∗ F Now let us define g . For p ∈ g ⊆ j(Q[κ,λ] ), define a new partial function p∗ with dom(p∗ ) = dom(p) by letting p∗ be equal to p unless p contradicts j H, in which case we flip the appropriate bits so that p∗ is compatible with the elements of j H. More precisely, working in V [G][H], let Q be the partial function
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with dom(Q) ⊆ dom(j(QF j H. Given p ∈ g, let p∗ be the condition in j(QF [κ,λ] )) defined by Q = [κ,λ] ) obtained from p by altering p on dom(p) ∩ dom(Q) so that p∗ agrees with Q. Let g ∗ = p∗ p ∈ g .
Clearly, j H ⊆ g ∗ , and it remains to argue that g ∗ remains an M [j(G)]-generic filter for j(QF [κ,λ] ). Since conditions p ∈ QF can have size greater than the critical point of j, it follows that j(p) need not [κ,λ] ∗ equal j p. Thus, some of the modifications we made in obtaining g occurred off of the range of j on the ghost coordinates, and so Woodin’s key lemma does not apply. We will use the following lemma to show that g ∗ remains an M [j(G)]-generic filter. M [j(G)] j(λ). Then Lemma 4 (New key lemma). Suppose B ∈ M [j(G)] with B ⊆ j(dom(QF [κ,λ] )) and |B| the set
IB = dom j(q) ∩ B q ∈ QF [κ,λ]
has size at most λ in V [G][H]. Proof. Let B be as in the statement of the lemma and let B = j(h)(j λ, α) where h : (Pκ λ)V × κ → V [G] Pλ+ (dom(QF , α < F (λ), and h ∈ V [G]. Then ran(h) is a subset of dom(QF [κ,λ] )) [κ,λ] ) in V [G] with V [G] <λ | ran(h)| λ. Since V [G] |= λ = λ by our GCH assumption, it will suffice to show that V [G] IB ⊆ j(d) ∩ B d ∈ Pλ ran(h) .
. We will show that dom(j(q)) ∩ B = j(d) ∩ B for some Suppose dom(j(q)) ∩ B ∈ IB where q ∈ QF [κ,λ] d ∈ Pλ ( ran(h))V [G] . Let d := dom(q) ∩ ran(h), then dom(j(q)) ∩ B = j(d) ∩ B since j(d) = dom j(q) ∩ ran j(h) ⊇ dom j(q) ∩ B.
2
∗ First let us show that the elements of g ∗ are conditions in j(QF [κ,λ] ) by showing that g ⊆ M [j(G)]. ∗ Suppose p ∈ g and let Idom(p) be the collection of all possible intersections of dom(p) with the domains of conditions j(q) where q ∈ H. By the genericity of H, it follows that H is a maximal filter on QF [κ,λ] and therefore,
Idom(p) = dom j(q) ∩ dom(p) q ∈ QF [κ,λ] . M [j(G)] < j(λ) and hence by Lemma 4, it follows that Since p is a condition in j(QF [κ,λ] ) we have | dom(p)| V [G][H] |= |Idom(p) | λ. Let Iα | α < λ ∈ V [G][H] be an enumeration of Idom(p) . For each α < λ choose a particular qα ∈ H with dom(j(qα )) ∩ dom(p) = Iα . Since M [j(G)] is closed under λ-sequences in F V [G][H], it follows that {j(qα ) | α < λ} is a directed subset of j(QF [κ,λ] ) in M [j(G)]. Notice that j(Q[κ,λ] ) is
IB := dom j(q) ∩ B q ∈ QF [κ,λ]
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satisfies V [G][H] |= |IB | λ. Let Eα | α < λ ∈ V [G][H] be an enumeration of IB . For each α < λ, choose qα ∈ H with dom(j(qα )) ∩ B = Eα . Choose rα ∈ g with dom(rα ) = dom(j(qα )). Define Δα := (δ, ξ, ζ) ∈ dom(rα ) rα (δ, ξ, ζ) = j(qα )(δ, ξ, ζ) .
Clearly, Δα ∈ M [j(G)] since rα , j(qα ) ∈ M [j(G)]. Then Δα | α < λ ∈ M [j(G)] because M [j(G)] is closed under λ-sequences in V [G][H][g0 ] by the λ-distributivity of the forcing adding g0 . Hence Δ :=
{Δα | α < λ}
F is in M [j(G)]. The automorphism πΔ : j(QF [κ,λ] ) → j(Q[κ,λ] ) that flips bits of conditions over Δ is in M [j(G)] −1 since Δ ∈ M [j(G)]. This implies that πΔ (A) is a maximal antichain of j(QF [κ,λ] ) in M [j(G)], and thus, by −1 the M [j(G)]-genericity of g, it follows that there is a condition r ∈ g∩πΔ (A). Furthermore, r∗ = πΔ (r) ∈ g ∗ and r∗ meets A. Hence g ∗ is M [j(G)]-generic. This shows that j lifts to j : V [G][H] → M [j(G)][j(H)] where j(G) = G ∗ H ∗ Gtail and j(H) = g ∗ in V [G][H][g0 ].
3.4. Lifting j through j0 (QF [κ,λ] ) By Claim 1 above, j0 (QF [κ,λ] ) is λ-distributive in V [G][H]. By an easy argument involving intersect ing λ open dense subsets of j0 (QF [κ,λ] ) (see [2, Proposition 15.1]), one can show that j g0 generates an F M [j(G)][j(H)]-generic filter for j(j0 (Q[κ,λ] )). Thus j lifts to j : V [G][H][g0 ] → M [j(G)][j(H)][j(g0 )] in V [G][H][g0 ] witnessing that κ is λ-supercompact in V [G][H][g0 ]. 2 4. A corollary and some open questions In this section we will discuss the extent to which our methods for proving Theorem 1 can be extended to cases in which κ is λ-supercompact where λ is a singular cardinal. First, we have an easy Corollary to Theorem 1. Corollary 3. Suppose GCH holds and there is an elementary embedding j : V → M with critical point κ such that for some singular cardinal λ > κ with cf(λ) < κ one has (1) M λ ⊆ M , (2) F : [κ, λ]∩REG → CARD is a function satisfying the requirements of Easton’s theorem, and (3) j(κ) > sup{F (γ) | γ ∈ [κ, λ) ∩ REG}+ . Then there is a cofinality-preserving forcing extension in which κ remains λ-supercompact and for every regular cardinal γ ∈ [κ, λ] one has 2γ = F (γ). One can see that Corollary 3 follows directly from Theorem 1 by extending the function F to F˜ with dom(F˜ ) = [κ, λ+ ] ∩ REG as follows. If F is eventually constant on [κ, λ) ∩ REG then let F˜ (λ+ ) be this constant value and F˜ [κ, λ) ∩ REG = F . It follows that the embedding j in the hypothesis of Corollary 3 witnesses that κ is λ<κ = λ+ -supercompact, and thus one can apply Theorem 1 to j and F˜ in order to obtain a forcing extension in which κ is λ-supercompact and 2γ = F (γ) for all γ ∈ [κ, λ) ∩ REG. In fact, in this case, the hypothesis in Corollary 3(3) can be weakened to j(κ) > sup{F (γ) | γ ∈ [κ, λ) ∩ REG}. For the remaining case in which F is not eventually constant, we may extend F to F˜ with dom(F˜ ) = [κ, λ+ ] ∩ REG by letting F˜ (λ+ ) = sup{F (γ) | γ ∈ [κ, λ) ∩ REG}+ and then apply Theorem 1. This suggests the following natural question, which our methods do not seem to answer. Question 2. Suppose GCH holds and κ is λ-supercompact where λ is a singular cardinal with cf(λ) κ and F : [κ, λ) ∩ REG → CARD is a function such that ∀α, β ∈ dom(F ) α < cf(F (α)) and α < β =⇒
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F (α) F (β). Suppose the λ-supercompactness of κ is witnessed by j : V → M and j(κ) > sup{F (γ) | γ ∈ [κ, λ) ∩ REG}+ . Is there a cofinality-preserving forcing extension preserving the λ-supercompactness of κ in which 2γ = F (γ) for all regular cardinals γ ∈ [κ, λ) ∩ REG? Under the assumption of Question 2 we have λ<κ = λ. Hence, it seems unlikely that there is an easy argument, similar to the above argument for Corollary 3, that would answer Question 2. A more promising strategy for answering Question 2 is to improve Lemma 4 to allow for the surgery argument to be carried out. The main obstacles to carrying out this strategy seem to be that if λ is singular then conditions in the F ++ -c.c. Easton-support forcing QF [κ,λ) can have support cofinal in λ and Q[κ,λ) is merely λ We close with another natural question, drawing inspiration from [4], regarding controlling the behavior of the continuum function globally, not just on [κ, λ] ∩ REG, while preserving the λ-supercompactness of κ. Notice that in the proof of Theorem 1, the forcing iteration up to κ is defined in terms of the function f : κ → Vκ , which was added by forcing such that j(f )(κ) = F (see Lemma 3). Indeed, in the final model of Theorem 1, the behavior of the continuum function below κ is dictated by the function f , and hence the methods of this paper do not provide an answer to the following. Question 3. Suppose GCH holds, F : REG → CARD is a function satisfying Easton’s requirements, and κ is λ-supercompact witnessed by j : V → M where j(κ) > F (λ). What additional assumptions will allow one to force the continuum function to agree with F at every regular cardinal while preserving the λ-supercompactness of κ? References [1] Brent Cody, The failure of GCH at a degree of supercompactness, MLQ Math. Log. Q. 58 (1–2) (2012) 83–94. [2] James Cummings, Iterated forcing and elementary embeddings, in: Akihiro Kanamori, Matthew Foreman (Eds.), Handbook of Set Theory, vol. 2, Springer, 2010, pp. 775–883, Chapter 14. [3] William B. Easton, Powers of regular cardinals, Ann. Math. Logic 1 (2) (1970) 139–178. [4] Sy-David Friedman, Radek Honzik, Easton’s theorem and large cardinals, Ann. Pure Appl. Logic 154 (3) (2008) 191–208. [5] Sy-David Friedman, Radek Honzik, Supercompactness and failures of GCH, Fund. Math. 219 (1) (2012) 15–36. [6] Joel David Hamkins, The lottery preparation, Ann. Pure Appl. Logic 101 (2–3) (2000) 103–146. [7] Joel David Hamkins, Tall cardinals, MLQ Math. Log. Q. 55 (1) (2009) 68–86. [8] Thomas Jech, Set Theory: The Third Millennium Edition, Revised and Expanded, Springer, 2003. [9] Richard Laver, Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel J. Math. 29 (4) (1978) 385–388. [10] Richard Laver, Certain very large cardinals are not created in small forcing extensions, Ann. Pure Appl. Logic 149 (1–3) (2007) 1–6. [11] Telis K. Menas, Consistency results concerning supercompactness, Trans. Amer. Math. Soc. 223 (1976) 61–91.
Contents lists available at ScienceDirect
Annals of Pure and Applied Logic www.elsevier.com/locate/apal
On supercompactness and the continuum function ✩ Brent Cody a,∗,1 , Menachem Magidor b,2 a The Fields Institute for Research in Mathematical Sciences, 222 College Street, Toronto, Ontario M5S 2N2, Canada b Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
a r t i c l e
i n f o
Article history: Received 20 May 2013 Received in revised form 10 September 2013 Accepted 10 September 2013 Available online 7 October 2013 MSC: 03E05 03E10 03E35 03E40 03E50 03E55
a b s t r a c t Given a cardinal κ that is λ-supercompact for some regular cardinal λ κ and assuming GCH, we show that one can force the continuum function to agree with any function F : [κ, λ] ∩ REG → CARD satisfying ∀α, β ∈ dom(F ) α < cf(F (α)) and α < β =⇒ F (α) F (β), while preserving the λ-supercompactness of κ from a hypothesis that is of the weakest possible consistency strength, namely, from the hypothesis that there is an elementary embedding j : V → M with critical point κ such that M λ ⊆ M and j(κ) > F (λ). Our argument extends Woodin’s technique of surgically modifying a generic filter to a new case: Woodin’s key lemma applies when modifications are done on the range of j, whereas our argument uses a new key lemma to handle modifications done off of the range of j on the ghost coordinates. This work answers a question of Friedman and Honzik [5]. We also discuss several related open questions. © 2013 Elsevier B.V. All rights reserved.
Keywords: Supercompact cardinal Continuum function Forcing Large cardinal
1. Introduction The behavior of the continuum function γ → 2γ on the regular cardinals was shown, by Easton, to be highly undetermined by the axioms of ZFC. Easton proved [3] that if F is any function from the regular cardinals to the cardinals satisfying α < cf(F (α)) and α < β =⇒ F (α) F (β), then there is a cofinality-preserving forcing extension in which 2γ = F (γ) for every regular cardinal γ. Large cardinal axioms impose additional restrictions on the continuum function on the regular cardinals. For example, ✩ The first author would like to thank the Fields Institute as well as the University of Prince Edward Island for their support while this work was being carried out. * Corresponding author. E-mail addresses: [email protected] (B. Cody), [email protected] (M. Magidor). URL: http://www.people.vcu.edu/~bmcody/ (B. Cody). 1 Current address: Virginia Commonwealth University, Department of Mathematics and Applied Mathematics, 1015 Floyd Avenue, Richmond, Virginia, 23284, United States. 2 Tel.: +972 2 65 84143.
0168-0072/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.apal.2013.09.001
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if κ is a supercompact cardinal and GCH holds below κ, then GCH holds everywhere. It therefore seems natural to address the question: What functions can be forced to coincide with the continuum function on the regular cardinals while preserving large cardinals? From what hypotheses? In particular, let us consider the following question. Question 1. Given a λ-supercompact cardinal κ where λ κ is a regular cardinal, and assuming GCH, what functions F from the regular cardinals to the cardinals can be forced to equal the continuum function on the interval [κ, λ] while preserving the λ-supercompactness of κ and preserving cardinals? From what hypotheses? Menas [11] proved that, assuming GCH, one can force the continuum function to agree at every regular cardinal with any locally definable 3 function F satisfying the requirements of Easton’s theorem, while preserving all cofinalities and preserving the supercompactness of a cardinal κ. If κ is a Laver-indestructible supercompact cardinal in a model V , as in [9], it easily follows that one can force over this model to achieve any reasonable behavior of the continuum function at and above κ, while preserving the supercompactness of κ. In particular, starting with a Laver-indestructible supercompact cardinal, one can obtain a model with a measurable cardinal at which GCH fails. However, one can also obtain a model with a measurable cardinal at which GCH fails from a much weaker large cardinal assumption. Woodin proved that the existence of a measurable cardinal at which GCH fails is equiconsistent with the existence of an elementary embedding j : V → M with critical point κ such that M κ ⊆ M and j(κ) > κ++ (see [2, Theorem 25.1]). Woodin’s argument illustrates that under certain conditions, one may perform a type of surgical modification on a generic filter g to obtain g ∗ in order to meet the lifting criterion, j G ⊆ g ∗ , and such that g ∗ remains generic. In Woodin’s proof, the modifications made to g in order to obtain g ∗ only occur on the range of j, and his key lemma shows that such changes are relatively mild in the sense that for a given condition p ∈ g, the set over which modifications are made to obtain p∗ ∈ g ∗ has size at most κ. Hamkins showed [7] that Woodin’s method could be applied to obtain an indestructibility theorem for tall cardinals. The first author extended Woodin’s surgery method to the case of partially supercompact cardinals in [1]. It is shown in [1] that the existence of a λ-supercompact cardinal κ such that 2λ λ++ is equiconsistent with the following hypothesis. (∗) There is an elementary embedding j : V → M with critical point κ such that M λ ⊆ M and j(κ) > λ++ . The method used in [1] is to, after a suitable preparatory iteration, blow up the size of the powerset of κ using Cohen forcing, in order to achieve 2κ = λ++ and then use Woodin’s method of surgery to lift the elementary embedding. Thus, one obtains a model in which κ is λ-supercompact and GCH fails at λ, because 2κ = λ++ . Answering a question posed in [1], Friedman and Honzik [5] used a variant of Sacks forcing for uncountable cardinals to show, from the hypothesis (∗), one can obtain a forcing extension in which κ is λ-supercompact, GCH holds on [κ, λ), and 2λ λ++ . The methods of both [1] and [5] leave open the following question, which appears in [5]. Assuming GCH and (∗), where κ < γ < λ are regular cardinals, is there a cofinality-preserving forcing extension in which κ remains λ-supercompact, GCH holds on the interval [κ, γ), and 2γ = λ++ ? In this article we answer this question, and indeed, provide a full answer to Question 1, by proving the following theorem. Theorem 1. Suppose GCH holds, κ < λ are regular cardinals, and F : [κ, λ] ∩ REG → CARD is a function such that ∀α, β ∈ dom(F ) α < cf(F (α)) and α < β =⇒ F (α) F (β). If there is an elementary embedding 3 A function F is locally definable if there is a sentence ψ, true in V , and a formula ϕ(x, y) such that for all cardinals γ, if Hγ |= ψ, then F has a closure point at γ and for all α, β < γ, we have F (α) = β ↔ Hγ |= ϕ(α, β).
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j : V → M with critical point κ such that M λ ⊆ M and j(κ) > F (λ), then there is a cofinality-preserving forcing extension in which κ remains λ-supercompact and 2γ = F (γ) for every regular cardinal γ ∈ [κ, λ]. The stage κ forcing Q in our proof will be an Easton-support product forcing for adding F (γ)-subsets to γ for each regular cardinal γ ∈ [κ, λ]. Since conditions p in this forcing will have size greater than the critical point of the embedding j under consideration, it follows that j(p) = j p and hence j(p) will have ghost-coordinates. Part of the modification we will perform on a certain generic filter g to obtain g ∗ with j H ⊆ g ∗ will occur off of the range of j on these ghost coordinates. Hence, Woodin’s key lemma does not apply. We will formulate and prove a new key lemma (Lemma 4 below) that allows much of the rest of the argument to be carried out as before. Essentially, our new key lemma shows that the set over which modifications are made to a condition p ∈ g to obtain p∗ ∈ g ∗ can be broken up into λ pieces, each of which is in the relevant model. The hypothesis in Theorem 1 is of optimal consistency strength for the simple reason that if j : V → M witnesses that κ is λ-supercompact, then M < j(κ). 2 λ 2λ
In Section 4 we discuss an easy corollary of Theorem 1 and an open question both addressing the case when λ is a singular cardinal. 2. Preliminaries 2.1. Background material We assume familiarity with the large cardinal notions of measurable and supercompact cardinals as well as with the characterization of these notions in terms of the existence of nontrivial elementary embeddings from the universe V into transitive inner models M ⊆ V . Some familiarity with the techniques of lifting large cardinal embeddings to forcing extensions is also assumed (see [2, Sections 8 and 9] and [8, Chapter 21]). 2.2. Supercompactness with tallness We say that a cardinal κ is θ-tall if θ > κ is an ordinal and there is an elementary embedding j : V → M with critical point κ such that j(κ) > θ and M κ ⊆ M . We say that κ is λ-supercompact with tallness θ if κ λ are cardinals, θ > λ is an ordinal, and there is a j : V → M with critical point κ such that M λ ⊆ M and j(κ) > θ. Such cardinals have been studied by Hamkins [7], Cody [1], Friedman and Honzik [5], and others. The next lemma appears in [1], and is easy to verify by factoring an elementary embedding through the ultrapower by an extender. Lemma 1. If κ is λ-supercompact with tallness θ then there is an embedding j : V → M witnessing this such that M = j(h) j λ, α | h : Pκ λ × κ → V & α < θλ & h ∈ V .
2.3. Easton’s product forcing Let us quickly review Easton’s product forcing and fix some notation. Suppose κ and λ are regular cardinals and F : [κ, λ] ∩ REG → CARD is a function satisfying Easton’s requirements; that is, for all regular cardinals α, β ∈ [κ, λ] one has
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α < cf F (α) and
(E1)
α < β =⇒ F (α) F (β).
(E2)
γ Let QF [κ,λ] denote the Easton-support product of Cohen forcing to achieve 2 = F (γ) for every regular F γ ∈ [κ, λ], by forcing over a model of GCH. We can regard conditions p ∈ Q[κ,λ] as functions satisfying the following.
• Every element in dom(p) is of the form (γ, α, β) where γ ∈ [κ, λ] is a regular cardinal, α < γ, and β < F (γ). • (Easton support) For each regular cardinal γ ∈ [κ, λ] we have (δ, α, β) ∈ dom(p) δ γ < γ.
• ran(p) ⊆ {0, 1}. Let dom(QF [κ,λ] ) :=
{dom(p) | p ∈ QF [κ,λ] } and notice that
dom QF [κ,λ] = (δ, α, β) δ ∈ [κ, λ] ∩ REG, α < δ, and β < F (δ)
By [3] we obtain the following. γ Lemma 2. Assuming GCH, forcing with the poset QF [κ,λ] preserves all cofinalities and achieves 2 = F (γ) for every regular cardinal γ ∈ [κ, λ] while preserving GCH otherwise.
One may also see [8, Theorem 15.18] for further details regarding the partial order QF [κ,λ] and Lemma 2. 3. Proof of the main theorem The next lemma will allow us to define the forcing iteration we will use to prove Theorem 1. Notice that if GCH holds and κ, λ, F , and j are as in the statement of Theorem 1 then F (λ)λ = F (λ) since λ < cf(F (λ)). Thus, under the hypothesis of Theorem 1, it follows from Lemma 1 that we may assume without loss of generality that j : V → M is such that M = j(h) j λ, α h : Pκ λ × κ → V & α < F (λ) & h ∈ V .
(3.1)
Lemma 3. Assume GCH and that κ, λ, F , and j : V → M are as in the hypothesis of Theorem 1. Furthermore, assume that M is as in (3.1). Then there is a cofinality-preserving forcing extension V [G] such that the following hold. (1) GCH holds in V [G]. (2) There is a partial function f from κ to Vκ in V [G] such that j lifts to j : V [G] → M [j(G)] and j(f )(κ) = F . (3) M [j(G)] = {j(h)(j λ, α) | h : (Pκ λ)V × κ → V & α < F (λ) & h ∈ V [G]}. M Proof. Fix x = xξ | ξ < κ , a well-ordering of Vκ . Then j( x) = zξ | ξ < j(κ) is a well-ordering of Vj(κ) . λ M Since M ⊆ M it is clear that F ∈ M and hence F ∈ Vj(κ) . Fix α < j(κ) such that F = zα . Let Fκ be Woodin’s poset for adding a partial function from κ to κ, defined as follows. Conditions in Fκ are partial functions p ⊆ κ × κ satisfying the following conditions.
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• γ ∈ dom(p) =⇒ γ < κ is an inaccessible cardinal and p γ ⊆ γ. • For every inaccessible cardinal μ < κ one has |p μ| < μ. The ordering on Fκ is defined by p p if and only if p ⊇ p . For a proof that under GCH the poset Fκ preserves all cofinalities and does not disturb the continuum function, see [6, Theorem 1.3]. Let G be V -generic for Fκ and let f˜ = G. Clearly f˜ is a partial function from κ to κ. Let us show that in M the poset j(Fκ ) factors below a condition p0 as j(Fκ )/p0 ∼ = Fκ × Ftail such that Ftail is F (λ)-closed in M . Notice that {(κ, α)} is a condition in Fκ and that the forcing j(Fκ ) can be factored below {(κ, α)} as j(Fκ )/{(κ, α)} ∼ = Fκ ×F[γ0 ,j(κ)) where γ0 denotes the least M -inaccessible cardinal greater than α, and F[γ0 ,j(κ)) is a γ0 -closed poset in M . If γ0 F (λ) then we can take p0 = {(κ, α)} and Ftail = F[γ0 ,j(κ)) . Assuming γ0 < F (λ), let p0 = {(κ, α), (γ0 , F (λ))} and factor j(Fκ ) below the condition p0 = {(κ, α), (γ0 , F (λ))} to obtain j(Fκ )/ (κ, α), γ0 , F (λ) ∼ = Fκ × F[γ1 ,j(κ))
where γ1 is the least M -inaccessible cardinal greater than F (λ) and F[γ1 ,j(κ)) is γ1 -closed in M . In this case we take Ftail = F[γ1 ,j(κ)) . Now we will show that there is an M -generic filter K for j(Fκ )/p0 ∼ = Fκ × Ftail in V [G] satisfying the lifting criterion j G ⊆ K, and hence j lifts to j : V [G] → M [j(G)] (see [2, Proposition 9.1]) where j(G) is the upward closure of K in j(Fκ ). This will suffice for (2) because p0 ∈ j(G) ensures that j(f˜)(κ) = α, and then working in V [G], one can define a function f with dom(f ) ⊆ κ by f (ξ) = xf˜(ξ) , and this function satisfies j(f )(κ) = zj(f˜)(κ) = zα = F . Let us show that K, as above, can be built in V [G]. First we show that there is an M -generic filter Gtail for Ftail in V . Let X denote the set of all dense subsets of all tails of the forcing Fκ . It follows that X has size at most 2κ = κ+ in V . Since M is as in (3.1), every dense subset D of Ftail in M is represented by a function h : Pκ λ × κ → X as D = j(h)(j λ, α) where α < F (λ) and h ∈ V . Since there are at most <κ (2κ )λ = (2κ )λ = 2λ = λ+ such functions we can enumerate them as hξ | ξ < λ+ ∈ V . Working in V we will build a decreasing sequence pξ | ξ < λ+ with pξ ∈ Ftail such that pξ meets every dense subset of Ftail in M represented by some hζ for ζ ξ. Assume pζ has been constructed for ζ < ξ. At stage ξ < λ, since M λ ⊆ M in V , we may let rξ ∈ Ftail be a lower bound of all previously constructed conditions. Since j(hξ )(j λ, α) | α < F (λ) ∈ M and Ftail is F (λ)-closed in M , it follows that there is a single condition pξ below rξ meeting every dense subset of Ftail in the sequence j(hξ )(j λ, α) | α < F (λ) . Since every dense subset of Ftail in M has a name that is represented by some function on our list hξ | ξ < λ+ , we can use pξ | ξ < λ+ to generate Gtail ∈ V an M -generic filter for Ftail . Now let K = G × Gtail . It is easy to verify that K is M -generic for j(Fκ )/p0 ∼ = Fκ × Ftail . We now show that (3) holds. If x ∈ M [j(G)] then x has a j(Fκ )-name, x˙ ∈ M such that x = x˙ j(G) . Thus we may write x˙ = j(h)(j λ, α) for some h : Pκ λ × κ → V , α < F (λ), and h ∈ V . Working in V [G] define ˜ : (Pκ λ)V × κ → V [G] by letting h(z, ˜ ξ) = h(z, ξ)G whenever h(z, ξ) is an Fκ -name and letting a function h ˜ ˜ ˜ h(z, ξ) = ∅ otherwise. Then h ∈ V [G] and j(h)(j λ, α) = j(h)(j λ, α)j(G) = x˙ j(G) = x. 2 Let us restate the main theorem. Theorem 2. Suppose GCH holds, κ < λ are regular cardinals, and F : [κ, λ] ∩ REG → CARD is a function such that ∀α, β ∈ dom(F ) α < cf(F (α)) and α < β =⇒ F (α) F (β). If there is an elementary embedding j : V → M with critical point κ such that M λ ⊆ M and j(κ) > F (λ), then there is a cofinality-preserving forcing extension in which κ remains λ-supercompact and 2γ = F (γ) for every regular cardinal γ ∈ [κ, λ]. be an Proof. To simplify notation later on, let V denote the model we start with and let j : V → M λ elementary embedding, which is a definable class of V such that (1) M ⊆ M , (2) F is as in the statement
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of the theorem, and (3) j(κ) > F (λ). Since GCH implies λ<κ = λ and F (λ)λ = F (λ), we can further assume without loss of generality that = M j(h) j λ, α h : Pκ λ × κ → V & α < F (λ) & h ∈ V . be the forcing extension of Lemma 3. So [ where Now let V = V [G] j lifts to j : V → M = M j(G)] M = j(h) j λ, α h : (Pκ λ)V × κ & α < F (λ) & h ∈ V ,
and there is a partial function f from κ to Vκ in V such that j(f )(κ) = F . Working in V , define Yf ⊆ dom(f ) to be the set of all γ < κ such that the following properties hold. (1) γ ∈ dom(f ) is an inaccessible cardinal and f γ ⊆ Vγ . (2) For some regular cardinal λγ ∈ [γ, κ), the value of f (γ) is a function from [γ, λγ ] ∩ REG to CARD satisfying the requirements of Easton’s theorem. Clearly, Yf must have measure one with respect to the normal measure U = {X ⊆ κ | κ ∈ j(X)}. ˙ γ ) | γ < κ + 1 be the length κ + 1 Easton-support iteration defined as follows. Let Pκ+1 = (Pγ , Q ˙γ =Q ˙ f (γ) be a Pγ -name for the Easton-support product Qf (γ) as defined in V Pγ • If γ ∈ Yf then let Q [γ,λγ ] [γ,λγ ] f (γ)
(see Section 2 above for a definition of the poset Q[γ,λγ ] ). ˙ γ is a Pγ -name for trivial forcing. • If γ < κ and γ ∈ / Yf , then Q F ˙κ =Q ˙F • For the stage κ forcing, let Q [κ,λ] be a Pκ -name for the Easton-support product Q[κ,λ] as defined Pκ in V . ˙ F . Standard arguments show that Pκ ∗ Q ˙ F preserves cofinalities and Let G ∗ H be V -generic for Pκ ∗ Q [κ,λ] [κ,λ] that in V [G][H] one has 2δ = F (δ) for every regular cardinal δ in [κ, λ]. Our argument follows Woodin’s: we show that there is a further cofinality-preserving forcing extension V [G][H][g0 ] in which κ remains λ-supercompact, and the desired continuum function is preserved on the interval [κ, λ]. 3.1. Lifting the embedding through Pκ ˙ j(f )(κ) ∗ P˙ tail . Since j(f )(κ) = F By elementarity, the forcing j(Pκ ) can be factored in M as j(Pκ ) ∼ = Pκ ∗ Q [κ,λκ ] by Lemma 3(3), and since M λ ⊆ M , it follows that ˙ F ∗ P˙ tail . j(Pκ ) ∼ = Pκ ∗ Q [κ,λ] Since the next stage of nontrivial forcing in j(Pκ ) beyond κ must be beyond F (λ), it follows that Ptail := (P˙ tail )G∗H is F (λ)-closed in M [G][H]. Now we will show that there is an M [G][H]-generic filter for Ptail in V [G][H]. The argument is similar to that in the proof of Lemma 3. For ξ < κ let Xξ denote the collection of all nice Pξ -names for dense subsets of the tail P˙ ξ,κ of the iteration Pκ . It follows that Xξ has size at most 2κ and hence X := {Xξ | ξ < κ} has ˙ F -name for D. size at most 2κ . Suppose D ∈ M [G][H] is a dense subset of Ptail and let D˙ ∈ M be a Pκ ∗ Q [κ,λ] ˙ Without loss of generality we may assume that D ∈ j(X) and hence there is a function h : (P λ)V ×κ → X ˙ D
κ
in V with D˙ = j(h)(j λ, α) for some α < F (λ). In other words, every dense subset of Ptail in M [G][H] ˙ F -name that is represented by a function (Pκ λ)V × κ → X in V . Since there are at most has a Pκ ∗ Q [κ,λ]
(2κ )λ
<κ
= 2λ = λ+ -such functions we can enumerate them as hξ | ξ < λ+ ∈ V . Working in V [G][H] we
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build a decreasing sequence pξ | ξ < λ+ with pξ ∈ Ptail meeting every dense subset of Ptail in M [G][H] as follows. Assume pζ has been constructed for ζ < ξ. At stage ξ < λ, since M [G][H]λ ⊆ M [G][H] in V [G][H], we may let rξ ∈ Ptail be a lower bound of all previously constructed conditions. Since j(hξ )(j λ, α)G∗H | α < F (λ) ∈ M [G][H] and Ptail is F (λ)-closed in M [G][H], it follows that there is a single condition pξ below rξ meeting every dense subset of Ptail in the sequence j(hξ )(j λ, α)G∗H | α < F (λ) . Since every dense subset of Ptail in M [G][H] has a name that represented by some function on our list hξ | ξ < λ+ , we can use pξ | ξ < λ+ to generate an M [G][H]-generic filter, call it Gtail ∈ V [G][H]. Since conditions in Pκ have bounded support, it follows that j G ⊆ G ∗ H ∗ Gtail , and thus we may lift the embedding to j : V [G] → M [j(G)] in V [G][H] where j(G) = G∗H ∗Gtail (as before, see [2, Proposition 9.1]). We have
M j(G) = j(h) j λ, α h : (Pκ λ)V × κ → V [G] & α < F (λ) & h ∈ V [G] .
3.2. Obtaining an M [j(G)]-generic filter for j(QF [κ,λ] ) We will factor the embedding j : V [G] → M [j(G)] through an ultrapower embedding j0 , force over F V [G][H] with j0 (QF [κ,λ] ) to obtain g0 , and then transfer g0 to an M [j(G)]-generic filter g for j(Q[κ,λ] ). F We will also show that forcing with j0 (Q[κ,λ] ) over V [G][H] preserves cardinals and does not disturb the continuum function below, or at, λ. Let X = {j(h)(j λ) | h : (Pκ λ)V → V [G] where h ∈ V [G]}. Then it follows that X ≺ M [j(G)]. Let k : M0 → M [j(G)] be the inverse of the Mostowski collapse π : X → M0 and let j0 : V [G] → M0 be defined by j0 := π ◦ j. It follows that j0 is the ultrapower embedding by the measure U0 := {X ⊆ (Pκ λ)V | j λ ∈ j(X)} where U0 ∈ V [G][H]. It is shown in [10] that the ground model is definable in any set forcing extension. Thus, by elementarity, M0 is of the form M0 [j0 (G)], where M0 ⊆ M0 and j0 (G) ⊆ j0 (Pκ ) ∈ M0 0 is M0 -generic. Furthermore, j0 (G) = G ∗ H0 ∗ G0tail where H0 is M0 [G]-generic for π(QF [κ,λ] ) and Gtail is M [G][H0 ]-generic for the tail of the iteration j0 (Pκ ) above stage κ. The following diagram is commutative. V [G]
j
j0
M [j(G)] k
M0 [j0 (G)] It follows that j0 is a class of V [G][H0 ], M0 [j0 (G)] is closed under λ-sequences in V [G][H0 ], and that j0 (κ) > π(F (λ)) with |j0 (κ)|V = λ+ . Hence the cardinal structures of M0 [j0 (G)] and V [G][H0 ] are identical up to and including λ+ . Furthermore, since j λ ∈ X it follows that X λ ⊆ X in V [G][H] which implies λ+ ⊆ X and crit(k) λ+ . Notice that F ∈ dom(π) = X because F = j(f )(κ) = k(j0 (f ))(κ) = k(j0 (f )(κ)) ∈ ran(k) = π(F ) dom(π). Thus π(QF [κ,λ] ) = Q[κ,λ] , where π(F ) is a function with domain π([κ, λ]) = [κ, λ] satisfying the requirements of Easton’s theorem in M0 [G]. Let g0 be V [G][H]-generic for j0 (QF [κ,λ] ). F The next claim shows that forcing with j0 (Q[κ,λ] ) over V [G][H] preserves cardinals and does not disturb the continuum function below, or at, λ. ++ -c.c. in V [G][H]. Claim 1. j0 (QF [κ,λ] ) is λ-distributive and λ
Proof. First we will demonstrate the distributivity. By elementarity, j0 (QF [κ,λ] ) is λ-closed in M0 [j0 (G)]. Since G0tail ∈ V [G][H0 ], it follows that j0 (QF ) ∈ V [G][H ] and since M [j 0 0 0 (G)] is closed under λ-sequences [κ,λ] F in V [G][H0 ], it follows that j0 (Q[κ,λ] ) is λ-closed in V [G][H0 ]. Notice that H0 is V [G]-generic for π(QF [κ,λ] ) = π(F ) V [G] F + and that the quotient forcing Q[κ,λ] /H0 is λ -c.c. in V [G][H0 ] Q[κ,λ] = γ∈[κ,λ]∩REG Add(γ, π(F (γ)))
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∗
∗ since it is isomorphic to QF [κ,λ] in that model, where F is a function with domain REG ∩ [κ, λ] such that ∗ + ∗ F (μ) = 0 if F (μ) λ and F (μ) = F (μ) otherwise. Easton’s Lemma states that λ-closed forcing remains λ-distributive after λ+ -c.c. forcing. Hence j0 (QF [κ,λ] ) is λ-distributive in V [G][H]. F ++ Now let us show that j0 (Q[κ,λ] ) is λ -c.c. in V [G][H]. Each p ∈ j0 (QF [κ,λ] ) can be written as p = V F j (h )(j λ) for some function h : (P λ) → Q with h ∈ V [G]. For each p ∈ j (QF ) the domain 0
p
0
p
κ
[κ,λ]
p
0
[κ,λ]
of hp has size λ in V [G], and thus hp leads to hp : λ → QF [κ,λ] , which can be viewed as a condition in the F full support product of λ copies of Q[κ,λ] taken in V [G]. Let us denote this product by QF [κ,λ] . We will show ++ F ++ that j0 (QF ) is λ -c.c. in V [G][H] by arguing that Q is λ -c.c. in V [G][H] and that an antichain [κ,λ] [κ,λ] F ++ F ++ in V [G][H] would lead to an antichain of Q[κ,λ] of size λ in V [G][H]. of j0 (Q[κ,λ] ) of size λ F ++ We now briefly describe how to prove that Q[κ,λ] is λ -c.c. in V [G][H]. An easy delta-system argument ++ shows that QF -c.c. in V [G]. Suppose A is an antichain of size δ of QF [κ,λ] is λ [κ,λ] in V [G][H]. We will show F F F ∼ that A leads to an antichain of Q[κ,λ] = Q[κ,λ] × Q[κ,λ] in V [G] of size δ. Let ˙ ˙ q A˙ is an antichain of QF [κ,λ] and f : δ → A is bijective ˙ ˙ where q ∈ QF ˇα where pα ∈ QF [κ,λ] and AH = A. For each α < δ let qα q be such that qα f (α) = p [κ,λ] . F F F We have Q[κ,λ] ∼ = Q[κ,λ] × Q[κ,λ] in V [G], and it is easy to check that W := {(qα , pα ) | α < δ} is an antichain F F ++ of QF -c.c. in V [G][H]. [κ,λ] × Q[κ,λ] in V [G] of size δ. This shows that Q[κ,λ] is λ It remains to show that an antichain of j0 (QF ) in V [G][H] with size λ++ would lead to an antichain [κ,λ] F ++ of Q[κ,λ] in V [G][H] of size λ . Suppose A is an antichain of j0 (QF [κ,λ] ) with size δ in V [G][H]. Each p ∈ A ˜
is of the form j0 (hp )(j λ) where hp : (Pκ λ)V → QF [κ,λ] . As mentioned above, each hp leads to a condition F hp ∈ Q[κ,λ] . It is easy to check that A := {hp | p ∈ A} is an antichain of QF [κ,λ] . 2 Let us now show that g0 can be transferred along k to an M [j(G)]-generic filter for j(QF [κ,λ] ). F Claim 2. k g0 ⊆ j(QF [κ,λ] ) generates an M [j(G)]-generic filter g for j(Q[κ,λ] ). Proof. Suppose D ∈ M [j(G)] is an open dense subset of j(QF [κ,λ] ) and let D = j(h)(j λ, α) for some h ∈ V [G] with dom(h) = (Pκ λ)V × κ and α < F (λ). Without loss of generality, let us assume that every element of the range of h is a dense subset of QF [κ,λ] in V [G]. We have D = j(h)(j λ, α) = k(j0 (h))(j λ, α). Since π(F (λ)) < j0 (κ) we may define a function h ∈ M0 [j0 (G)] with dom( h) = π(F (λ)) by h(ξ) = j0 (h)(j0 λ, ξ). Then dom(k( h)) = k(π(F (λ))) = F (λ) and since the critical point of k is greater than h)(α) = k(j0 (h))(k(j λ), α) = j(h)(j λ, α) = D. Now the range of h is a collection of λ we have k( 0
F π(F (λ)) open dense subsets of j0 (QF [κ,λ] ). Since j0 (Q[κ,λ] ) is π(F (λ))-distributive in M0 [j0 (G)], one sees
= ran( h) is an open dense subset of j0 (QF that D [κ,λ] ). Hence there is a condition p ∈ g0 ∩ D and by ⊆ D. 2 elementarity, k(p) ∈ k g0 ∩ k(D)
3.3. Performing surgery ∗ ∗ With the M [j(G)]-generic filter g for j(QF [κ,λ] ) in hand, we surgically modify g to obtain g with j H ⊆ g , and then argue that g ∗ remains an M [j(G)]-generic filter for j(QF [κ,λ] ). ∗ F Now let us define g . For p ∈ g ⊆ j(Q[κ,λ] ), define a new partial function p∗ with dom(p∗ ) = dom(p) by letting p∗ be equal to p unless p contradicts j H, in which case we flip the appropriate bits so that p∗ is compatible with the elements of j H. More precisely, working in V [G][H], let Q be the partial function
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with dom(Q) ⊆ dom(j(QF j H. Given p ∈ g, let p∗ be the condition in j(QF [κ,λ] )) defined by Q = [κ,λ] ) obtained from p by altering p on dom(p) ∩ dom(Q) so that p∗ agrees with Q. Let g ∗ = p∗ p ∈ g .
Clearly, j H ⊆ g ∗ , and it remains to argue that g ∗ remains an M [j(G)]-generic filter for j(QF [κ,λ] ). Since conditions p ∈ QF can have size greater than the critical point of j, it follows that j(p) need not [κ,λ] ∗ equal j p. Thus, some of the modifications we made in obtaining g occurred off of the range of j on the ghost coordinates, and so Woodin’s key lemma does not apply. We will use the following lemma to show that g ∗ remains an M [j(G)]-generic filter. M [j(G)] j(λ). Then Lemma 4 (New key lemma). Suppose B ∈ M [j(G)] with B ⊆ j(dom(QF [κ,λ] )) and |B| the set
IB = dom j(q) ∩ B q ∈ QF [κ,λ]
has size at most λ in V [G][H]. Proof. Let B be as in the statement of the lemma and let B = j(h)(j λ, α) where h : (Pκ λ)V × κ → V [G] Pλ+ (dom(QF , α < F (λ), and h ∈ V [G]. Then ran(h) is a subset of dom(QF [κ,λ] )) [κ,λ] ) in V [G] with V [G] <λ | ran(h)| λ. Since V [G] |= λ = λ by our GCH assumption, it will suffice to show that V [G] IB ⊆ j(d) ∩ B d ∈ Pλ ran(h) .
. We will show that dom(j(q)) ∩ B = j(d) ∩ B for some Suppose dom(j(q)) ∩ B ∈ IB where q ∈ QF [κ,λ] d ∈ Pλ ( ran(h))V [G] . Let d := dom(q) ∩ ran(h), then dom(j(q)) ∩ B = j(d) ∩ B since j(d) = dom j(q) ∩ ran j(h) ⊇ dom j(q) ∩ B.
2
∗ First let us show that the elements of g ∗ are conditions in j(QF [κ,λ] ) by showing that g ⊆ M [j(G)]. ∗ Suppose p ∈ g and let Idom(p) be the collection of all possible intersections of dom(p) with the domains of conditions j(q) where q ∈ H. By the genericity of H, it follows that H is a maximal filter on QF [κ,λ] and therefore,
Idom(p) = dom j(q) ∩ dom(p) q ∈ QF [κ,λ] . M [j(G)] < j(λ) and hence by Lemma 4, it follows that Since p is a condition in j(QF [κ,λ] ) we have | dom(p)| V [G][H] |= |Idom(p) | λ. Let Iα | α < λ ∈ V [G][H] be an enumeration of Idom(p) . For each α < λ choose a particular qα ∈ H with dom(j(qα )) ∩ dom(p) = Iα . Since M [j(G)] is closed under λ-sequences in F V [G][H], it follows that {j(qα ) | α < λ} is a directed subset of j(QF [κ,λ] ) in M [j(G)]. Notice that j(Q[κ,λ] ) is
IB := dom j(q) ∩ B q ∈ QF [κ,λ]
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satisfies V [G][H] |= |IB | λ. Let Eα | α < λ ∈ V [G][H] be an enumeration of IB . For each α < λ, choose qα ∈ H with dom(j(qα )) ∩ B = Eα . Choose rα ∈ g with dom(rα ) = dom(j(qα )). Define Δα := (δ, ξ, ζ) ∈ dom(rα ) rα (δ, ξ, ζ) = j(qα )(δ, ξ, ζ) .
Clearly, Δα ∈ M [j(G)] since rα , j(qα ) ∈ M [j(G)]. Then Δα | α < λ ∈ M [j(G)] because M [j(G)] is closed under λ-sequences in V [G][H][g0 ] by the λ-distributivity of the forcing adding g0 . Hence Δ :=
{Δα | α < λ}
F is in M [j(G)]. The automorphism πΔ : j(QF [κ,λ] ) → j(Q[κ,λ] ) that flips bits of conditions over Δ is in M [j(G)] −1 since Δ ∈ M [j(G)]. This implies that πΔ (A) is a maximal antichain of j(QF [κ,λ] ) in M [j(G)], and thus, by −1 the M [j(G)]-genericity of g, it follows that there is a condition r ∈ g∩πΔ (A). Furthermore, r∗ = πΔ (r) ∈ g ∗ and r∗ meets A. Hence g ∗ is M [j(G)]-generic. This shows that j lifts to j : V [G][H] → M [j(G)][j(H)] where j(G) = G ∗ H ∗ Gtail and j(H) = g ∗ in V [G][H][g0 ].
3.4. Lifting j through j0 (QF [κ,λ] ) By Claim 1 above, j0 (QF [κ,λ] ) is λ-distributive in V [G][H]. By an easy argument involving intersect ing λ open dense subsets of j0 (QF [κ,λ] ) (see [2, Proposition 15.1]), one can show that j g0 generates an F M [j(G)][j(H)]-generic filter for j(j0 (Q[κ,λ] )). Thus j lifts to j : V [G][H][g0 ] → M [j(G)][j(H)][j(g0 )] in V [G][H][g0 ] witnessing that κ is λ-supercompact in V [G][H][g0 ]. 2 4. A corollary and some open questions In this section we will discuss the extent to which our methods for proving Theorem 1 can be extended to cases in which κ is λ-supercompact where λ is a singular cardinal. First, we have an easy Corollary to Theorem 1. Corollary 3. Suppose GCH holds and there is an elementary embedding j : V → M with critical point κ such that for some singular cardinal λ > κ with cf(λ) < κ one has (1) M λ ⊆ M , (2) F : [κ, λ]∩REG → CARD is a function satisfying the requirements of Easton’s theorem, and (3) j(κ) > sup{F (γ) | γ ∈ [κ, λ) ∩ REG}+ . Then there is a cofinality-preserving forcing extension in which κ remains λ-supercompact and for every regular cardinal γ ∈ [κ, λ] one has 2γ = F (γ). One can see that Corollary 3 follows directly from Theorem 1 by extending the function F to F˜ with dom(F˜ ) = [κ, λ+ ] ∩ REG as follows. If F is eventually constant on [κ, λ) ∩ REG then let F˜ (λ+ ) be this constant value and F˜ [κ, λ) ∩ REG = F . It follows that the embedding j in the hypothesis of Corollary 3 witnesses that κ is λ<κ = λ+ -supercompact, and thus one can apply Theorem 1 to j and F˜ in order to obtain a forcing extension in which κ is λ-supercompact and 2γ = F (γ) for all γ ∈ [κ, λ) ∩ REG. In fact, in this case, the hypothesis in Corollary 3(3) can be weakened to j(κ) > sup{F (γ) | γ ∈ [κ, λ) ∩ REG}. For the remaining case in which F is not eventually constant, we may extend F to F˜ with dom(F˜ ) = [κ, λ+ ] ∩ REG by letting F˜ (λ+ ) = sup{F (γ) | γ ∈ [κ, λ) ∩ REG}+ and then apply Theorem 1. This suggests the following natural question, which our methods do not seem to answer. Question 2. Suppose GCH holds and κ is λ-supercompact where λ is a singular cardinal with cf(λ) κ and F : [κ, λ) ∩ REG → CARD is a function such that ∀α, β ∈ dom(F ) α < cf(F (α)) and α < β =⇒
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F (α) F (β). Suppose the λ-supercompactness of κ is witnessed by j : V → M and j(κ) > sup{F (γ) | γ ∈ [κ, λ) ∩ REG}+ . Is there a cofinality-preserving forcing extension preserving the λ-supercompactness of κ in which 2γ = F (γ) for all regular cardinals γ ∈ [κ, λ) ∩ REG? Under the assumption of Question 2 we have λ<κ = λ. Hence, it seems unlikely that there is an easy argument, similar to the above argument for Corollary 3, that would answer Question 2. A more promising strategy for answering Question 2 is to improve Lemma 4 to allow for the surgery argument to be carried out. The main obstacles to carrying out this strategy seem to be that if λ is singular then conditions in the F ++ -c.c. Easton-support forcing QF [κ,λ) can have support cofinal in λ and Q[κ,λ) is merely λ We close with another natural question, drawing inspiration from [4], regarding controlling the behavior of the continuum function globally, not just on [κ, λ] ∩ REG, while preserving the λ-supercompactness of κ. Notice that in the proof of Theorem 1, the forcing iteration up to κ is defined in terms of the function f : κ → Vκ , which was added by forcing such that j(f )(κ) = F (see Lemma 3). Indeed, in the final model of Theorem 1, the behavior of the continuum function below κ is dictated by the function f , and hence the methods of this paper do not provide an answer to the following. Question 3. Suppose GCH holds, F : REG → CARD is a function satisfying Easton’s requirements, and κ is λ-supercompact witnessed by j : V → M where j(κ) > F (λ). What additional assumptions will allow one to force the continuum function to agree with F at every regular cardinal while preserving the λ-supercompactness of κ? References [1] Brent Cody, The failure of GCH at a degree of supercompactness, MLQ Math. Log. Q. 58 (1–2) (2012) 83–94. [2] James Cummings, Iterated forcing and elementary embeddings, in: Akihiro Kanamori, Matthew Foreman (Eds.), Handbook of Set Theory, vol. 2, Springer, 2010, pp. 775–883, Chapter 14. [3] William B. Easton, Powers of regular cardinals, Ann. Math. Logic 1 (2) (1970) 139–178. [4] Sy-David Friedman, Radek Honzik, Easton’s theorem and large cardinals, Ann. Pure Appl. Logic 154 (3) (2008) 191–208. [5] Sy-David Friedman, Radek Honzik, Supercompactness and failures of GCH, Fund. Math. 219 (1) (2012) 15–36. [6] Joel David Hamkins, The lottery preparation, Ann. Pure Appl. Logic 101 (2–3) (2000) 103–146. [7] Joel David Hamkins, Tall cardinals, MLQ Math. Log. Q. 55 (1) (2009) 68–86. [8] Thomas Jech, Set Theory: The Third Millennium Edition, Revised and Expanded, Springer, 2003. [9] Richard Laver, Making the supercompactness of κ indestructible under κ-directed closed forcing, Israel J. Math. 29 (4) (1978) 385–388. [10] Richard Laver, Certain very large cardinals are not created in small forcing extensions, Ann. Pure Appl. Logic 149 (1–3) (2007) 1–6. [11] Telis K. Menas, Consistency results concerning supercompactness, Trans. Amer. Math. Soc. 223 (1976) 61–91.